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and let D stand for not to be not…good' and be placed under A。 Then
either A or B will belong to everything； but they will never belong to
the same thing； and either C or D will belong to everything； but
they will never belong to the same thing。 And B must belong to
everything to which C belongs。 For if it is true to say 'it is a
not…white'； it is true also to say 'it is not white'： for it is
impossible that a thing should simultaneously be white and be
not…white； or be a not…white log and be a white log； consequently if
the affirmation does not belong； the denial must belong。 But C does
not always belong to B： for what is not a log at all； cannot be a
not…white log either。 On the other hand D belongs to everything to
which A belongs。 For either C or D belongs to everything to which A
belongs。 But since a thing cannot be simultaneously not…white and
white； D must belong to everything to which A belongs。 For of that
which is white it is true to say that it is not not…white。 But A is
not true of all D。 For of that which is not a log at all it is not
true to say A； viz。 that it is a white log。 Consequently D is true；
but A is not true； i。e。 that it is a white log。 It is clear also
that A and C cannot together belong to the same thing； and that B
and D may possibly belong to the same thing。
Privative terms are similarly related positive ter terms respect
of this arrangement。 Let A stand for 'equal'； B for 'not equal'； C for
'unequal'； D for 'not unequal'。
In many things also； to some of which something belongs which does
not belong to others； the negation may be true in a similar way；
viz。 that all are not white or that each is not white； while that each
is not…white or all are not…white is false。 Similarly also 'every
animal is not…white' is not the negation of 'every animal is white'
（for both are false）： the proper negation is 'every animal is not
white'。 Since it is clear that 'it is not…white' and 'it is not white'
mean different things； and one is an affirmation； the other a
denial； it is evident that the method of proving each cannot be the
same； e。g。 that whatever is an animal is not white or may not be
white； and that it is true to call it not…white； for this means that
it is not…white。 But we may prove that it is true to call it white
or not…white in the same way for both are proved constructively by
means of the first figure。 For the expression 'it is true' stands on a
similar footing to 'it is'。 For the negation of 'it is true to call it
white' is not 'it is true to call it not…white' but 'it is not true to
call it white'。 If then it is to be true to say that whatever is a man
is musical or is not…musical； we must assume that whatever is an
animal either is musical or is not…musical； and the proof has been
made。 That whatever is a man is not musical is proved destructively in
the three ways mentioned。
In general whenever A and B are such that they cannot belong at
the same time to the same thing； and one of the two necessarily
belongs to everything； and again C and D are related in the same
way； and A follows C but the relation cannot be reversed； then D
must follow B and the relation cannot be reversed。 And A and D may
belong to the same thing； but B and C cannot。 First it is clear from
the following consideration that D follows B。 For since either C or
D necessarily belongs to everything； and since C cannot belong to that
to which B belongs； because it carries A along with it and A and B
cannot belong to the same thing； it is clear that D must follow B。
Again since C does not reciprocate with but A； but C or D belongs to
everything； it is possible that A and D should belong to the same
thing。 But B and C cannot belong to the same thing； because A
follows C； and so something impossible results。 It is clear then
that B does not reciprocate with D either； since it is possible that D
and A should belong at the same time to the same thing。
It results sometimes even in such an arrangement of terms that one
is deceived through not apprehending the opposites rightly； one of
which must belong to everything； e。g。 we may reason that 'if A and B
cannot belong at the same time to the same thing； but it is
necessary that one of them should belong to whatever the other does
not belong to： and again C and D are related in the same way； and
follows everything which C follows： it will result that B belongs
necessarily to everything to which D belongs'： but this is false。
'Assume that F stands for the negation of A and B； and again that H
stands for the negation of C and D。 It is necessary then that either A
or F should belong to everything： for either the affirmation or the
denial must belong。 And again either C or H must belong to everything：
for they are related as affirmation and denial。 And ex hypothesi A
belongs to everything ever thing to which C belongs。 Therefore H
belongs to everything to which F belongs。 Again since either F or B
belongs to everything； and similarly either H or D； and since H
follows F； B must follow D： for we know this。 If then A follows C； B
must follow D'。 But this is false： for as we proved the sequence is
reversed in terms so constituted。 The fallacy arises because perhaps
it is not necessary that A or F should belong to everything； or that F
or B should belong to everything： for F is not the denial of A。 For
not good is the negation of good： and not…good is not identical with
'neither good nor not…good'。 Similarly also with C and D。 For two
negations have been assumed in respect to one term。
Book II
1
WE have already explained the number of the figures； the character
and number of the premisses； when and how a syllogism is formed；
further what we must look for when a refuting and establishing
propositions； and how we should investigate a given problem in any
branch of inquiry； also by what means we shall obtain principles
appropriate to each subject。 Since some syllogisms are universal；
others particular； all the universal syllogisms give more than one
result； and of particular syllogisms the affirmative yield more than
one； the negative yield only the stated conclusion。 For all
propositions are convertible save only the particular negative： and
the conclusion states one definite thing about another definite thing。
Consequently all syllogisms save the particular negative yield more
than one conclusion； e。g。 if A has been proved to to all or to some B；
then B must belong to some A： and if A has been proved to belong to no